Define coefficient of restitution.

Hint: The term coefficient of restitution needs to be defined, go through the context of the term and define the term in detail, consider various cases for coefficient of restitution as well.

Complete step by step answer:
Understand that, the ratio of the difference between the final velocities of colliding objects to the difference between the initial velocities is known as coefficient of restitution.
Write the expression of the coefficient of restitution \[e\]
\[e = \dfrac{{{v_2} - {v_1}}}{{{u_1} - {u_2}}}\]
Here, \[{v_2} - {v_1}\] is the velocity of separation, \[{u_1} - {u_2}\] is the velocity of approach, \[{u_1}\] is the initial velocity of the object 1, \[{u_2}\] is the initial velocity of the object 2, \[{v_1}\] is the final velocity of the object 1 and \[{v_2}\] is the initial velocity of the object 2.
Consider the elastic collision: The velocity of separation becomes equal to the velocity of approach, Therefore the difference between the final velocities is equal to difference between the initial velocities.
Write the coefficient of restitution for elastic collision,
\[e = \dfrac{{{v_2} - {v_1}}}{{{u_1} - {u_2}}}\]
Substitute \[{u_1} - {u_2}\] for \[{v_2} - {v_1}\]
\[\
e = \dfrac{{{u_1} - {u_2}}}{{{u_1} - {u_2}}} \\
 = 1 \\
\ \]
Consider the inelastic collision: The velocity of separation is less than the velocity of approach, therefore, the difference between the final velocities is always less difference between the initial velocities. And velocity of separation is not zero.
Write the coefficient of restitution for inelastic collision,
\[e = \dfrac{{{v_2} - {v_1}}}{{{u_1} - {u_2}}} < 1\]
Consider the Perfectly inelastic collision: The velocity of separation becomes zero, after the collision both objects move with the same velocity.
Write the coefficient of restitution for elastic collision,
\[e = \dfrac{{{v_2} - {v_1}}}{{{u_1} - {u_2}}}\]
Substitute \[{v_2}\] equals to \[{v_1}\]
\[\
e = \dfrac{{{v_1} - {v_1}}}{{{u_1} - {u_2}}} \\
 = 0 \\
\ \]

Note:
The coefficient of the restitution is defined using the context behind the term.
The coefficient of restitution helps in identifying the types of collisions between two objects.